It sounds as though DNS isn't working.
Try
nslookup hp.com from a command prompt on your SME box and see what happens
Then try
nslookup hp.com 4.2.2.1 (which will do the same lookup, but try to get the answer from '4.2.2.1', which works from every network I've tried it from in the last 10 years...)
Look at
/var/log/tinydns/current to see if you're getting DNS error messages on your SME.
Browsing problems could be caused by the squid proxy, but this wouldn't cause any problems w/ email.
By default, your SME server is trying to do DNS lookups directly from the Internet's Root servers - if your DlinkG604T(10.1.1.1) is blocking or proxying DNS requests this might not work - so if the 'nslookup' tests don't work correctly you might look for a DNS proxy setting in your router.
Lastly, I had the same behavior a few months ago and it turned out to be due to a full hard drive. Tinydns would fail miserably at every lookup as it tried to update its cache.
Try
df -k and check the 'Use%' on the device mounted on "/".